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Time consistency of dynamic risk measures in markets with transaction costs

  • Zachary Feinstein
  • Birgit Rudloff
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    The paper concerns primal and dual representations as well as time consistency of set-valued dynamic risk measures. Set-valued risk measures appear naturally when markets with transaction costs are considered and capital requirements can be made in a basket of currencies or assets. Time consistency of scalar risk measures can be generalized to set-valued risk measures in different ways. The most intuitive generalization is called time consistency. We will show that the equivalence between a recursive form of the risk measure and time consistency, which is a central result in the scalar case, does not hold in the set-valued framework. Instead, we propose an alternative generalization, which we will call multi-portfolio time consistency and show in the main result of the paper that this property is indeed equivalent to the recursive form as well as to an additive property for the acceptance sets. Multi-portfolio time consistency is a stronger property than time consistency. In the scalar case, both notions coincide.

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    File URL: http://arxiv.org/pdf/1201.1483
    File Function: Latest version
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    Paper provided by arXiv.org in its series Papers with number 1201.1483.

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    Date of creation: Jan 2012
    Date of revision: Dec 2012
    Publication status: Published in Quantitative Finance 13 (9), 1473-1489, (2013)
    Handle: RePEc:arx:papers:1201.1483
    Contact details of provider: Web page: http://arxiv.org/

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    1. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Conditional Risk Mappings," Risk and Insurance 0404002, EconWPA, revised 08 Oct 2005.
    2. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    3. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and Dynamic Convex Risk Measures," SFB 649 Discussion Papers SFB649DP2005-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    5. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    6. Cheridito, Patrick & Stadje, Mitja, 2009. "Time-inconsistency of VaR and time-consistent alternatives," Finance Research Letters, Elsevier, vol. 6(1), pages 40-46, March.
    7. Elyès Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued Coherent Risk Measures," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00167154, HAL.
    8. Touzi, Nizar & Meddeb, Moncef & Jouini, Elyès, 2004. "Vector-valued Coherent Risk Measures," Economics Papers from University Paris Dauphine 123456789/353, Paris Dauphine University.
    9. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    10. Patrick Cheridito & Michael Kupper, 2011. "Composition Of Time-Consistent Dynamic Monetary Risk Measures In Discrete Time," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 137-162.
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